Hi

Is the inverse of a function always a function ?

I asked myself this question because of an exercise that I was solving but couldn't find an answer.

We have [tex]f(x)=\sqrt{1+sinx}[/tex] defined over [tex]\left [\frac{-\pi}{2} ,\frac{\pi}{2} \right ][/tex]

Let's see its curve on the graph and also the one of its inverse:

Screenshot_20180203-201741.jpg

The inverse's curve doesn't seem to be a function to me (maybe I'm missing some information in my mind).

A function is a map (every x has a unique y-value), while on the inverse's curve some x-values have 2 y-values.

By doing some calculations, I get [tex]f^{-1}(x)\,=\,\arcsin(x^{2}-1)[/tex]

Its curve (the one I calculated) is like that (the grey one below):

Screenshot_20180203-201851.jpg

The part defined over negative x of this curve shocked me.

Can someone explain to me how this can happen ?

And why shouldn't the grey curve be like the purple one ?

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